QUADRATIC EQUATIONS

JEE ADVANCED

Match the Following 1] Consider the quadratic equation + 2(2 + 1) + 9 5 = 0 List - I List-II

A. a > 7 1.Imaginary roots

B. a < 0 2.Negative roots

C. 2 < a < 5 3.One positive and one negative root

(a) A - 3 B - 1 C - 2 (b) A - 3 B 2 C - 1

(c) A - 2 B - 1 C - 3 (d) A - 1 B - 2 C - 3

2] Let a < b < c < d. Match with the column in which equation has root in

List - I List-II

A. (x a) (x c) + 2007 (x b) (x d) = 0 1.(c, d)

B. 2007 (x a)(x c) 2006 (x b)(x d) = 0 2.(a, b)

C. (x a)(x b) (x c) + (x d) = 0 3. ( , )a

D. (x a)(x b) + (x c) = 0 4. (b, c)

(a) A - 2 B - 3 C - 1 D - 4 (b) A - 3 B - 2 C - 1 D - 4

(c) A - 2 B - 1 C - 3 D - 4 (d) A - 1 B - 2 C - 3 D - 4

3] If and be the roots of the equation + + = 0 then

is equal to

(a) (b) , (c)

(d) none

4] The coefficient of in the quadratic equation + + = 0 was taken as 17 in place of 13, its roots were found to be 2 and 15. Then the original roots of the equation are

(a) 10, 5 (b) 10, 3 (c) 10, 3 (d) 10, 3

5] Two candidates attempt to solve a quadratic equation of the form + + = 0. One starts with a wrong value of p and finds the roots to be 2 and 6. The other starts with a

wrong value of q and finds the roots to be 2 and 9. Then the correct roots are

(a) 3, 4 (b) 3, 4 (c) 3, 4 (d) 3, 4

6] The expression + + , 0, , , is of one sign for all if and only if 4 < 0. However if 4 = 0, then + + is of one sign, except at =

, where the value of the expression is zero. In the remaining case + +

takes all types of values (i.e., negative, zero and positive).

If 1 + is a root of the equation () + + = 0; , {0}, < then (a) () < 0 for (, ) only

(b) () < 0 for (,) (,) only (c) () > 0 for all

(d) () < 0 for all

7] The expression + + , 0, , , is of one sign for all if and only if 4 < 0. However if 4 = 0, then + + is of one sign, except at =

, where the value of the expression is zero. In the remaining case + +

takes all types of values (i.e., negative, zero and positive)

If () + + 1; , , does not have linear factors over R, then (a) .() < 0 for all (b) () < 0 for all (c) 4 > 0 (d) () > 0 8] The expression + + , 0, , , is of one sign for all if and only if

4 < 0. However if 4 = 0, then + + is of one sign, except at =

, where the value of the expression is zero. In the remaining case + +

takes all types of values (i.e., negative, zero and positive)

If the equation () + + 1 = 0 does not have two distinct real roots, then + 1 is

(a) nonnegative

(b) nonpositive

(c) zero

(d) data is not sufficient to decide the sign of + 1

9] Solve the following for real values of :

i) 3| 4 + 2| = 5 4 ii) |( + 3)|( + 1) + |2 + 5| = 0

iii) 2|| |2 1| = 2 + 1 a) = 2 or 5; =

; 1 or = 3

b) = 2 or 5; =

; 1 or = 3 c) = 2 or 5; =

; 1 or = 3

d) None of these

10] Solve the following equations / in equations for real x:

i)

()() < () ii) log|| 2. log( 2) 1

iii)

0

a) (1,); < 7,5 < 2, 4; 2,2 15 b) (1,); < 7,5 < 2, 4; 2,2 15 c) (1,); > 7,5 < 2, 4; 2,215

d) None of these

11] Solve the equations for

= 1: + + = 2 a) 4, 14 b) 4, 14 c) 4,14 d) + 4, 14

12] If , be the roots of the equation, ( ) + 2 + 3 = 0 and , be the two values of for which and are connected by the relation,

+

=

, then find the quadratic

equation whose roots are

and

b) If are the roots of + + = 0 & , are the roots of + + = 0, show that , are the roots of

+

+ +

+

= 0

a) 3 68 18 = 0 b) 3 + 68 + 18 = 0 c) 3 68 + 18 = 0 d) 3 + 68 18 = 0

13] Find the range of values of a, such that () = ()

is always negative.

a) , b) ,

c) ,

d) ,

14] If , be the roots of the equation 3 + = 0 & , be those of the equation 12 + = 0 and ,, , are in G.P. Find A and B.

a) = 2 18, = 32288 b) = 2 18, = 32 288 c) = 218, = 32 288 d) = 218, = 32 288

15] Find the values of a for which 3 < () < 2 is valid for all real x. a) 2 < < 1 b) 2 < < 1 c) 2 < < 1 d) 2 < < 1 16] ( 5) + 4 = 0 be a quadratic equation. Find the value of a for which atleast

one root is positive.

a) (, 0] [25, ) b) (, 0] [25, )

c) (, 0] [25, ) d) (, 0] [25, )

17] ( 5) + 4 = 0 be a quadratic equation. Find the value of a for which one root is smaller than 2, the other root is greater than 2

a) [25, ) b) [-25, ) c) [-25,-] d) none

18] ( 5) + 4 = 0 be a quadratic equation. Find the value of a for which both

roots are greater than 2

a) (7, 1] b) (7, 1] c) (7, 1] d) none

19] ( 5) + 4 = 0 be a quadratic equation. Find the value of a for which both roots are smaller than 2

a) (7, 2) b) (7, 2) c) (7, 2) d) None

20] ( 5) + 4 = 0 be a quadratic equation. Find the value of a for which exactly

one of the roots lie in the interval (1, 2)

a) f b) y c) x d) O

21] ( 5) + 4 = 0 be a quadratic equation. Find the value of a for which both roots lie in the interval (1, 2)

a) (7, 2) b) (7, 2) c) (7, 2) d) none

22] ( 5) + 4 = 0 be a quadratic equation. Find the value of a for which atleast one root lie in the interval (1, 2)

a) (, 7) [25, ) b) (, 7) [25, )

c) (, 7) [25, ) d) none

] ( 5) + 4 = 0 be a quadratic equation. Find the value of a for which one root is greater than 2, the other roots is smaller than 1

a) (, 7) [25, ) b) (, 7) [25, )

c) (, 7) [25, ) d) none

24] If , are the two distinct roots of + 2( 3). + 9 = 0, then find the values of k such that , (6,1).

a) 6, b) 6,

c) 6,

d) none

25] Find all numbers a for each for which the least value of quadratic trinomial 4 4 + 2 + 2 on the interval 0 2 is equal to 3. a) = 1 + 2, 5 + 10 b) = 1 2, 5 10 c) = 1 2, 5 + 10 d) none